arXiv subjects

L. Beshaj

Publications and source records attributed to L. Beshaj.

4 recordsLinked to original sources

arXiv2019

Rational points in the moduli space of genus two

We build a database of genus 2 curves defined over $\mathbb Q$ which contains all curves with minimal absolute height $h \leq 5$, all curves with moduli height $\mathfrak h \leq 20$, and all curves with extra automorphisms in standard form $y^2=f(x^2)$ defined over $\mathbb Q$ with height $h \leq 101$. For each isomorphism class in the database, an equation over its minimal field of definition is provided, the automorphism group of the curve, Clebsch and Igusa invariants. The distribution of rational points in the moduli space $\mathcal M_2$ for which the field of moduli is a field of definition is discussed and some open problems are presented.

math.AG↗

arXiv2017

Minimal Weierstrass equations for genus 2 curves

We study the minimal Weierstrass equations for genus 2 curves defined over a ring of integers $\mathcal O_{\mathbb F}$. This is done via reduction theory and Julia invariant of binary sextics. We show that when the binary sextics has extra automorphisms this is usually easier to compute. Moreover, we show that when the curve is given in the standard form $y^2=f(x^2)$, where $f(x)$ is a monic polynomial, $f(0)=1$ which is defined over $\mathcal O_{\mathbb F}$ then this form is reduced.

math.AG↗

arXiv2015

Theta functions of superelliptic curves

In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components of the moduli space in terms of such functions.

math.CV↗

arXiv2015

Heights on algebraic curves

In these lectures we cover basics of the theory of heights starting with the heights in the projective space, heights of polynomials, and heights of the algebraic curves. We define the minimal height of binary forms and moduli height for algebraic curves and prove that the moduli height of superelliptic curves $\mathcal H (f) \leq c_0 \tilde H (f)$ where $c_0$ is a constant and $\tilde H$ the minimal height of the corresponding binary form. For genus $g=2$ and 3 such constant is explicitly determined. Furthermore, complete lists of curves of genus 2 and genus 3 hyperelliptic curves with height 1 are computed.

math.NT↗